Ultracompleteness of hyperspaces of compact sets

Abstract

A space X is called ultracomplete if it has countable character in its Stone–Cech compactification βX. A space X is called almost locally compact if the set of all points at which X is not locally compact is contained in a compact set of countable outer character. For a given Tychonoff space X let 2X be the hyperspace of all nonempty compact subsets of X endowed with the Vietoris topology. We prove that 2X is almost locally compact if and only if X is locally compact. We also prove that for a countably compact ultracomplete space X the hyperspace Fn(X)={K∈2X∣K has at most n points} is also countably compact ultracomplete for every natural number n. We also analyse ultracompleteness of Fn(X) and 2X.